| (1) | (\rho \,c_{pt})_m \frac{\partial T}{\partial t} + \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T - \nabla (\lambda_t \nabla T) - \ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial p_\alpha}{\partial t} + \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \nabla p = \frac{\delta E_H}{ \delta V \delta t} |
For a single-phase fluid flow this simplifies to
| (2) | \rho \, c_p \, \frac{\partial T}{\partial t} - {\bf \nabla}\, \left( \lambda \, {\bf \nabla} T \right) + \rho \, c_p \, {\bf u} \, {\bf \nabla} T - \phi \, \rho \ c_p \ \eta_s \ \frac{\partial p}{\partial t} + \rho \ c_p \ \epsilon \ {\bf u} {\bf \nabla} p = \frac{\delta E_H}{ \delta V \delta t} |
See also
Physics / Thermodynamics / Heat Transfer / Heat flow
[ Heat Flux ]